# Why do we need continuum approximation in fluid mechanics?

We know that a fluid in reality is not continuous. It has spaces and voids between atoms and molecules.

Continuum approximation is a famous approximation that is taken in any fluid mechanics textbook. It says that even though the fluid has spaces and voids it can be assumed to behave as a continuous media.

Why do we need to assume that a fluid is a continuous media? That is, what was the problem that we were facing when it was not continuous?

• because modeling a ten to the 23rd power of individual atoms isn't viable, nor even possible prior to supercomputers. Because mechanical engineering is mostly about things you can see. Commented Jul 28, 2022 at 14:13
• The same argument can be applied to integration & limit theorem. Despite finer slices, a point is eventually reached where the scale becomes molecular & atomic.
– Fred
Commented Jul 28, 2022 at 17:54

Materials were intuitively uniform for 60,000 years. A few people started guessing they might be "atomic" about 3000 years ago. They only became rigorously atomic about two hundred years ago. And they only got a rigorous continuum model about one hundred years ago. But they were being treated as such on an ad hoc basis long before then.

There isn't any conflict between the continuum model and the atomic viewpoint. There never was. The two developed in concert.

Boyle published his 1662 law that involved gas pressure, and they needed a way to measure and mathematically handle this rather poorly understood phenomena. The "elasticity" of a gas was a real dilemma. Boyle and Hooke imagined little springs between their imagined atoms.

So in the 17th C, you had a hypothesized atomic model whose behavior needed to agree with the measurements of the day, quantities we now associate with the continuum model.

Enter calculus, stage right, which was developed from little "infinitesimals" (generalized atoms.) The result was integral and differential calculus applied to continuous functions (in retrospect, this was an unfortunate choice of terms.) In order to harness the power of calculus, it helps to have a formal underpinning that allows you to treat pressure, density, velocity, and a host of other things you can measure as continuous functions. They didn't have that in the 18th C, but that didn't stop Bernoulli and Euler from applying calculus to fluids. Work, as defined by Coriolis (1826), didn't need calculus, just buckets of water and a rope. But there's only so much you can do with those, and not everyone has a mine shaft. A calculus-based definition of work was a lot more convenient.

So basically, calculus was a solution in search of a problem. Fluid dynamics was a reasonable candidate. After a century of ad hoc application and some decent successes, mathematicians and physicists went back and developed the formal underpinnings to justify what had been done. It let us consolidate thousands of ad hoc experiments into a few laws, and it allowed us to do performance-based design of dynamic systems like steam engines.

Burying the lead - You said "We know that a fluid in reality is not continuous. It has spaces and voids between atoms and molecules."

You are assuming the continuum model assumes a continuous structure. It doesn't. What the continuum model does is assume continuous function expressions that relate pressure, density, etc to each other. Continuous functions are actually defined based on the epislon-delta argument of Cauchy.

In his 1821 book Cours d'analyse, Cauchy discussed variable quantities, infinitesimals and limits, and defined continuity of $$y=f(x)$$ by saying that an infinitesimal change in x necessarily produces an infinitesimal change in y, while (Grabiner 1983) claims that he used a rigorous epsilon-delta definition in proofs.[2]

Continuum models are, and always have been, fully consistent with an atomic structure. They were produced with that structure in mind. It is the behavior of the atoms that has been captured in the continuum model.

• Prior to asking this question, I referred to a lot of books but nowhere was it explained as explicitly as you did. Thank You so much. Commented Jul 29, 2022 at 18:46
• You're right, I guess the whole issue with my understanding was that I was considering continuum treatment as treating fluid as a continuous medium (with no spaces). Some books that I referred do mention such an explanation (the spaces and voids one), but some of the famous books on Fluids Mechanics like Fox and McDonald discuss that continuum has to do with the continuity of functions (as you said). So, it is true that continuum doesn't mean fluid has become continuous, it means that properties are defined at every point and they're continuous. Commented Jul 29, 2022 at 18:46
• A question then arises in the study of Chaos, where infinitesimal changes in some variable result in very large consequences. With chaos then, can we conclude that Cauchy's assumption of continuous functions does not apply? Is it simply fortuitous that in non-chaotic systems continuous functions provide a good mathematical model to describe what happens in nature? Commented Jul 31, 2022 at 13:24
• Chaos requires lots of time. For short intervals of time, the system remains determinate, and, nevertheless, can be very smooth and well behaved on all trajectories over time. Chaos is a boundary value phenomena. It isn't apparent from inspecting the problem's configuration space. You have to ascertain how uncertainty propagates over time. Commented Jul 31, 2022 at 19:07

A continuum fluid model allows us to calculate things using average properties of the fluid at any point throughout its volume.

For the special situations where the molecular distance does become important, which is in the order of a billionth of a meter, the continuum model does not apply, and requires statistical techniques to study fluid flow

R C Hibbeler, Fluid Mechanics 2nd ed, 2018, Pearson, NY.

I believe this answers your question. Fluid on engineering have always been assumed to be continuum fluid until recently. Obviously turbulent flow with a gas mixed with liquid is different because that is a mixture of fluids with different properties.

The question is, when was fluid not treated as continuum model... which would be at molecular level study.